[RPG] apply Martial Arts Modified Weapon damage on Polearm Master Bonus attack

You make your attack with a quarterstaff using Str or Dex. You then deal damage either as per monk level or 1d6 for staff or 1d8 for versatile.

You then choose whether you make a monk bonus action attack or polearm bonus action attack, and get the replacement damage die that comes with the bonus action you chose.

As you gain levels only the monk attack damage would go up; your polearm bonus attack would remain at 1d4. You cannot apply your monk level bonus damage to the polearm bonus attack, but you would gain the bonuses of a magical weapon in that regard, both on attack and damage.

Sidenote: Some DMs may rule that using a quarterstaff in two-handed (versatile) mode doesn't count as a monk weapon, as two-handed weapons cannot be used for monk weapons. Such a ruling would remove the option to make an unarmed strike as a bonus action, but you would retain your polearm bonus strike option. For RAW the option remains.

Yes, your reach with the bonus attack is the same as your reach with the regular attack. The Reach weapon property states that:

This weapon adds 5 feet to your reach when you attack with it, as well as when determining your reach for opportunity attacks with it (see chapter 9).

So whenever you make an attack with a Reach weapon, your reach is 5 feet more than usual. There's no restrictions placed on how you get that attack. As long as you're using the right weapon, you get the extra reach.

In this specific case, you're attacking with the opposite end of the weapon, but it's still the same weapon. Your reach will be 10ft if it's a Reach weapon, or 5ft otherwise.